Summary Evaluation of reservoir parameters through well-test and decline-curve analysis is a current practice used to estimate formation parameters and to forecast production decline identifying different flow regimes, respectively. From practical experience, it has been observed that certain cases exhibit different wellbore pressure and production behavior from those presented in previous studies. The reason for this difference is not understood completely, but it can be found in the distribution of fractures within a naturally fractured reservoir (NFR). Currently, most of these reservoirs are studied by means of Euclidean models, which implicitly assume a uniform distribution of fractures and that all fractures are interconnected. However, evidence from outcrops, well logging, production-behavior studies, and the dynamic behavior observed in these systems, in general, indicate the above assumptions are not representative of these systems. Thus, the fractal theory can contribute to explain the above. The objective of this paper is to investigate the production-decline behavior in an NFR exhibiting single and double porosity with fractal networks of fractures. The diffusion equations used in this work are a fractal-continuity expression presented in previous studies in the literature and a more recent generalization of this equation, which includes a temporal fractional derivative. The second objective is to present a combined analysis methodology, which uses transient-well-test and boundary-dominated-decline production data to characterize an NFR exhibiting fractures, depending on scale. Several analytical solutions for different diffusion equations in fractal systems are presented in Laplace space for both constant-wellbore-pressure and pressure-variable-rate inner-boundary conditions. Both single- and dual-porosity systems are considered. For the case of single-porosity reservoirs, analytical solutions for different diffusion equations in fractal systems are presented. For the dual-porosity case, an approximate analytical solution, which uses a pseudosteady-state matrix-to-fractal fracture-transfer function, is introduced. This solution is compared with a finite-difference solution, and good agreement is found for both rate and cumulative production. Short- and long-time approximations are used to obtain practical procedures in time for determining some fractal parameters. Thus, this paper demonstrates the importance of analyzing both transient and boundary-dominated flow-rate data for a single-well situation to fully characterize an NFR exhibiting fractal geometry. Synthetic and field examples are presented to illustrate the methodology proposed in this work and to demonstrate that the fractal formulation consistently explains the peculiar behavior observed in some real production-decline curves. Introduction Evaluation of reservoir parameters through decline-curve analysis has become a common current practice (Fetkovich 1980; Fetkovich et al. 1987). The main objectives of the application of decline analysis are to estimate formation parameters and to forecast production decline by identifying different flow regimes. Different solutions have been proposed during both transient (Ehlig-Economides and Ramey 1981; Uraiet and Raghavan 1980) and boundary-dominated (Fetkovich 1980; Fetkovich et al. 1987; Ehlig-Economides and Ramey 1981; Arps 1945) flow periods. Both single- and double-porosity (Da Prat et al. 1981; Sageev et al. 1985) systems have been addressed. During the boundary-dominated-flow period, in homogeneous systems, there is a single production decline, but for NFRs in which the matrix participates, there are two decline periods, with an intermediate constant-flow period (Da Prat et al. 1981; Sageev et al. 1985). Carbonate reservoirs contain more than 60% of the world's remaining oil. Yet, the very nature of the rock makes these reservoirs unpredictable. Formations are heterogeneous, with irregular flow paths and circulation traps. In spite of this complexity, at present, all studies on constant-bottomhole-pressure tests found in petroleum literature assume Euclidean or standard geometry is applicable to both single-porosity reservoirs and NFRs (Fetkovich 1980; Fetkovich et al. 1987; Ehlig-Economides and Ramey 1981; Uraiet and Raghavan 1980; Arps 1945; Da Prat et al. 1981; Sageev et al. 1985), even though real reservoirs exhibit a higher level of complexity. Specifically, natural fractures are heterogeneities that are present in carbonate reservoirs on a wide range of spatial scales. It is well known that flow distribution within the reservoir is controlled mostly by the distribution of fractures (i.e., geometrical complexity). There could be regions in the reservoir with clusters of fractures and others without the presence of fractures. The presence of fractures at different scales represents a relevant element of uncertainty in the construction of a reservoir model. Thus, highly heterogeneous media constitute the basic components of an NFR, so Euclidean flow models have appeared powerless in some of these cases. Alternatively, fractal theory provides a method to describe the complex network of fractures (Sahimi and Yortsos 1970). The power-law behavior of fracture-size distributions, characteristic of fractal systems, has been found by Laubach and Gale (2006) and Ortega et al. (2006). Distributions of attributes such as length, height, or aperture can frequently be expressed as power laws. Scaling analysis is important because it enables us to infer fracture attributes such as fracture strike, number of fracture sets, and fracture intensity for larger fractures from the analysis of microfractures found in oriented sidewall cores. This approach offers a method to overcome fracture-sampling limitations, with microfractures as proxies for related macrofractures in the same rock volume (Laubach and Gale 2006; Ortega et al. 2006). The first fractal model applied to pressure-transient analysis was presented by Chang and Yortsos (1990). Their model describes an NFR that has, at different scales, poor fracture connectivity and disorderly spatial distribution in a proper fashion. Acuña et al. (1995) applied this model and found the wellbore pressure is a power-law function of time. Flamenco-Lopez and Camacho-Velazquez (2003) demonstrated that to characterize a NFR fully with a fractal geometry, it is necessary to analyze both transient- and pseudosteady-state-flow well pressure tests or to determine the fractal-model parameters from porosity well logs or another type of source. Regarding the generation of fracture networks, Acuña et al. (1995) used a mathematical method for this purpose, while Philip et al. (2005) used a fracture-mechanics-based crack-growth simulator, instead of a purely stochastic method, for the same objective. In spite of all the work done on decline-curve analysis, the problem of fully characterizing an NFR exhibiting fractal geometry by means of production data has not been addressed in the literature. Thus, the purpose of this work is to present analytical solutions during both transient- and boundary-dominated-flow periods and to show that it is possible to characterize an NFR having a fractal network of fractures with production-decline data.

TX 75083-3836, U.S.A., fax 01-972-952-9435. AbstractEvaluation of reservoir parameters through well test and decline curve analysis is a current practice used to estimate formation parameters and to forecast production decline identifying different flow regimes, respectively. From practical experience, it has been observed that certain cases exhibit different wellbore pressure and production behavior from those presented in previous studies. The reason for this difference is not understood completely but it can be found in the distribution of fractures within a Naturally Fractured Reservoir (NFR). Currently, most of these reservoirs are studied by means of Euclidean models, which implicitly assume a uniform distribution of fractures and that all fractures are interconnected.However, evidences from outcrops, well logging, and production behavior studies, and in general, the dynamic behavior observed in these systems, indicate that the above assumptions are not representative for these systems. Thus, the fractal theory can contribute to explain the above.Synthetic and field examples are presented to illustrate the methodology proposed in this work and to demonstrate that the fractal formulation explains consistently the peculiar behavior observed in some real production decline curves.

Summary This work presents a new approach to account for variable matrix-block size in the well performance of unconventional shale reservoirs. In contrast to the standard models that consider either a fixed matrix-block size or a stochastic distribution of sizes with no particular dependence on the spatial location, in this model we consider the case when the characteristic length of the blocks depends on the distance from the main hydraulic-fracture plane. In particular, we assume that, as a result of the hydraulic-fracturing treatment, the density of microfractures (natural and induced) is high near the hydraulic-fracture face, but gradually decreases away from it. In other words, the matrix-block size is an increasing function of the distance from the fracture face. We show that the boost of the contact area between the matrix blocks and the microfractures can be the determinant feature of hydrocarbon production from shale reservoirs, even when the natural and induced microfractures do not have uniform density throughout the whole stimulated reservoir volume. In addition, the elongated linear flow (the consensus characteristic signature in the well performance of shale systems) may be the result of an induced interporosity flow when the matrix-/fracture-permeability ratio is small. The duration of this flow regime increases if the effective size of the matrix-block distribution increases. On the other hand, the linear flow can be the result of the total-system response when the matrix-/fracture-permeability ratio is high; in this case, the beginning of the linear flow is strongly affected by the small matrix blocks near the hydraulic-fracture faces. The resulting mathematical model in Laplace space has a fundamentally different structure compared with standard dual-porosity models because of the spatial dependence of the parameters characterizing the interporosity flow. In this work, we develop the Airy-spline scheme, a new technique to calculate the pressure solution in an efficient way. Closed-form approximate formulae in the time domain are also provided, revealing that during the induced linear-interporosity-flow period, the production behavior is controlled by the logarithmic mean of the minimum and maximum matrix-block size. A field example from the Barnett shale is presented to illustrate the use of the new model.

Summary This work introduces a new model for the production-decline analysis (PDA) of hydraulically fractured wells on the basis of the concept of the induced permeability field. We consider the case when the hydraulic-fracturing operation—in addition to establishing the fundamental linear-flow geometry in the drainage volume—alters the ability of the formation to conduct fluids throughout, but with varying degrees depending on the distance from the main fracture plane. We show that, under these circumstances, the reservoir response departs from the uniform-permeability approach significantly. The new model differs from the once promising group of models that are inherently related to power-law-type variation of the permeability-area product and thus are burdened by a mathematical singularity inside the fracture. The analysis of field cases reveals that the induced permeability field can be properly represented by a linear or exponential function characterized by the maximal induced permeability k0 and the threshold permeability k*. Both these permeabilities are induced (superimposed on the formation) by the hydraulic-fracturing treatment; thus, the model can be considered as a simple, but nontrivial, formalization of the intuitive stimulated-reservoir-volume (SRV) concept. It is quite reasonable to assume that the maximum happens at the fracture face and that the minimum happens at the outer boundary of the SRV. The contrast between maximal and minimal permeability, SR = k0/k*, will be of considerable interest, and thus, we introduce a new term for it: stimulation ratio (SR). Knowledge of these parameters is crucial in evaluating the effectiveness of today's intensively stimulated well completions, especially multifractured horizontal wells in shale gas. The approach describes, in a straightforward manner, the production performance of such wells exhibiting transient linear flow and late-time boundary-dominated flow affected also by a skin effect (i.e., by an additional pressure drop in the system characterized by linear dependence on production rate). This work provides the induced-permeability-field model within the single-medium concept, and shows that some features widely believed to require a dual-medium (double-porosity) representation are already present. Advantages and drawbacks related to applying the concept in a dual-medium approach will be discussed in an upcoming work. We present the model and its analytical solution in Laplace space. We provide type curves for decline-curve analysis, closed-form approximate solutions in the time domain, field examples, and practical guidelines for the analysis of commonly occurring production characteristics of massively stimulated reservoirs.